Wireless communication standards such as 2nd generation Global System for Mobile Communications (GSM) based on time-division multiple access (TDMA) and/or frequency-division multiple access (FDMA), 3rd generation systems based on wideband code-division multiple access (WCDMA), and emerging 3rd Generation Partnership Program (3GPP) Long Term Evolution (LTE) systems based on orthogonal frequency-division multiplexing (OFDM) (in the downlink) and single-carrier FDMA (in the uplink), all employ multiple antennas to increase signal reception quality, and to enhance the system coverage. Although GSM systems primarily use multiple receiver antennas for link quality enhancement, 3G-WCDMA and LTE systems use multiple antennas both at the transmitter and the receiver.
Due to constructive and destructive addition of radio waves traveling over the air, the wireless channel between a transmitter antenna and a receiver antenna can be described as a frequency-selective and a time-varying random propagation medium. The frequency-selective nature of the channel produces multiple overlapping signals at the receiver, where each copy of the transmitted signal is attenuated by a random channel gain and delayed by a random time offset. A cyclic-prefix (CP) can be introduced to limit inter-symbol interference caused by channel frequency-selectivity. If CP length is larger than a channel's maximum delay spread, inter-symbol interference can be avoided. Additionally, relative motion between the transmitter and the receiver can introduce Doppler spread to the transmitted signal. For OFDM systems, inter-carrier interference can be avoided if the Doppler spread is smaller than the sub-carrier spacing.
Traditionally, channel estimation for multiple-antennas with OFDM modulation can be performed in two steps. In a first step, using pilot tones, a frequency-domain channel is estimated. In a second step, the frequency-domain channel is interpolated over the pilot tones of interest. This approach is known to yield satisfactory performance if the pilot density is high. With low pilot density, this approach suffers from interpolation errors (e.g., for tones between two given frequency tones), and extrapolation errors (e.g., for data tones beyond the vicinity of the pilot tones).